1. Dynamic Programming
Comp 122, Fall 2004

2. Longest Common Subsequence
Problem: Given 2 sequences, X = x1,...,xm and Y = y1,...,yn, find a common subsequence whose length is maximum.
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printing north carolina krzyzewski
Subsequence need not be consecutive, but must be in order.
Comp 122, Spring 2004

3. Other sequence questions
Edit distance: Given 2 sequences, X = x1,...,xm and Y = y1,...,yn, what is the minimum number of deletions, insertions, and changes that you must do to change one to another?
Protein sequence alignment: Given a score matrix on amino acid pairs, s(a,b) for a,b{}A, and 2 amino acid sequences, X = x1,...,xmAm and Y = y1,...,ynAn, find the alignment with lowest score…
Comp 122, Spring 2004

4. More problems
Optimal BST: Given sequence K = k1 < k2 <··· < kn of n sorted keys, with a search probability pi for each key ki, build a binary search tree (BST) with minimum expected search cost.
Matrix chain multiplication: Given a sequence of matrices A1 A2 … An, with Ai of dimension mini, insert parenthesis to minimize the total number of scalar multiplications.
Minimum convex decomposition of a polygon,
Hydrogen placement in protein structures, …
Comp 122, Spring 2004

5. Dynamic Programming
Dynamic Programming is an algorithm design technique for optimization problems: often minimizing or maximizing.
Like divide and conquer, DP solves problems by combining solutions to subproblems.
Unlike divide and conquer, subproblems are not independent.
Subproblems may share subsubproblems,
However, solution to one subproblem may not affect the solutions to other subproblems of the same problem. (More on this later.)
DP reduces computation by
Solving subproblems in a bottom-up fashion.
Storing solution to a subproblem the first time it is solved.
Looking up the solution when subproblem is encountered again.
Key: determine structure of optimal solutions
Comp 122, Spring 2004

6. Steps in Dynamic Programming
Characterize structure of an optimal solution.
Define value of optimal solution recursively.
Compute optimal solution values either top-down with caching or bottom-up in a table.
Construct an optimal solution from computed values.
We’ll study these with the help of examples.
Comp 122, Spring 2004

7. Longest Common Subsequence
Problem: Given 2 sequences, X = x1,...,xm and Y = y1,...,yn, find a common subsequence whose length is maximum.
springtime ncaa tournament basketball
printing north carolina snoeyink
Subsequence need not be consecutive, but must be in order.
Comp 122, Spring 2004

8. Problema
Problem: Given 2 sequences, X = x1,...,xm and Y = y1,...,yn, find a common subsequence whose length is maximum.
Let Z= z1,...,zk be the LCS of X and Y. Prove that is also LCS of Y and X.
Comp 122, Spring 2004

9. Naïve Algorithm
For every subsequence of X, check whether it’s a subsequence of Y .
Time: Θ(n2m).
2m subsequences of X to check.
Each subsequence takes Θ(n) time to check: scan Y for first letter, for second, and so on.
Comp 122, Spring 2004

10. Optimal Substructure
Theorem
Let Z = z1, , zk be any LCS of X and Y .
1. If xm = yn then zk = xm = yn and Zk-1 is an LCS of Xm-1 and Yn-1.
2. If xm  yn then either zk  xm and Z is an LCS of Xm-1 and Y .
or zk  yn and Z is an LCS of X and Yn-1.
Notation:
prefix Xi = x1,...,xi is the first i letters of X.
This says what any longest common subsequence must look like; do you believe it?
Comp 122, Spring 2004

11. Optimal Substructure Theorem
Let Z = z1, , zk be any LCS of X and Y .
1. If xm = yn, then zk = xm = yn and Zk-1 is an LCS of Xm-1 and Yn-1.
2. If xm  yn, then either zk  xm and Z is an LCS of Xm-1 and Y .
or zk  yn and Z is an LCS of X and Yn-1.
Proof: (case 1: xm = yn)
Any sequence Z’ that does not end in xm = yn can be made longer by adding xm = yn to the end. Therefore,
longest common subsequence (LCS) Z must end in xm = yn.
Zk-1 is a common subsequence of Xm-1 and Yn-1, and
there is no longer CS of Xm-1 and Yn-1, or Z would not be an LCS.
Comp 122, Spring 2004

12. Optimal Substructure Theorem
Let Z = z1, , zk be any LCS of X and Y .
1. If xm = yn, then zk = xm = yn and Zk-1 is an LCS of Xm-1 and Yn-1.
2. If xm  yn, then either zk  xm and Z is an LCS of Xm-1 and Y .
or zk  yn and Z is an LCS of X and Yn-1.
Proof: (case 2: xm  yn, and zk  xm)
Since Z does not end in xm,
Z is a common subsequence of Xm-1 and Y, and
there is no longer CS of Xm-1 and Y, or Z would not be an LCS.
Comp 122, Spring 2004

13. Recursive Solution Define c[i, j] = length of LCS of Xi and Yj .
We want c[m,n].
This gives a recursive algorithm and solves the problem. But does it solve it well?
Comp 122, Spring 2004

14. Recursive Solution c[springtime, printing]
c[springtim, printing] c[springtime, printin]
[springti, printing] [springtim, printin] [springtim, printin] [springtime, printi]
[springt, printing] [springti, printin] [springtim, printi] [springtime, print]
Comp 122, Spring 2004

15. Recursive Solution Keep track of c[a,b] in a table of nm entries:g s m e
Keep track of c[a,b] in a table of nm entries:
top/down
bottom/up
Comp 122, Spring 2004

16. Computing the length of an LCS
LCS-LENGTH (X, Y)
m ← length[X]
n ← length[Y]
for i ← 1 to m
do c[i, 0] ← 0
for j ← 0 to n
do c[0, j ] ← 0
do for j ← 1 to n
do if xi = yj
then c[i, j ] ← c[i1, j1] + 1
b[i, j ] ← “ ”
else if c[i1, j ] ≥ c[i, j1]
then c[i, j ] ← c[i 1, j ]
b[i, j ] ← “↑”
else c[i, j ] ← c[i, j1]
b[i, j ] ← “←”
return c and b
b[i, j ] points to table entry whose subproblem we used in solving LCS of Xi
and Yj.
c[m,n] contains the length of an LCS of X and Y.
Time: O(mn)
Comp 122, Spring 2004

17. Constructing an LCS PRINT-LCS (b, X, i, j) if i = 0 or j = 0
then return
if b[i, j ] = “ ”
then PRINT-LCS(b, X, i1, j1)
print xi
elseif b[i, j ] = “↑”
then PRINT-LCS(b, X, i1, j)
else PRINT-LCS(b, X, i, j1)
Initial call is PRINT-LCS (b, X,m, n).
When b[i, j ] = , we have extended LCS by one character. So LCS = entries with in them.
Time: O(m+n)
Comp 122, Spring 2004

18. Steps in Dynamic Programming
Characterize structure of an optimal solution.
Define value of optimal solution recursively.
Compute optimal solution values either top-down with caching or bottom-up in a table.
Construct an optimal solution from computed values.
We’ll study these with the help of examples.
Comp 122, Spring 2004

19. Optimal Binary Search Trees
Problem
Given sequence K = k1 < k2 <··· < kn of n sorted keys, with a search probability pi for each key ki.
Want to build a binary search tree (BST) with minimum expected search cost.
Actual cost = # of items examined.
For key ki, cost = depthT(ki)+1, where depthT(ki) = depth of ki in BST T .
Comp 122, Spring 2004

20. Expected Search Cost Sum of probabilities is 1. (15.16)
Comp 122, Spring 2004

21. Example
Consider 5 keys with these search probabilities: p1 = 0.25, p2 = 0.2, p3 = 0.05, p4 = 0.2, p5 = 0.3. k2
i depthT(ki) depthT(ki)·pi
1.15 k1 k4 k3 k5
Therefore, E[search cost] = 2.15.
Comp 122, Spring 2004

22. Example p1 = 0.25, p2 = 0.2, p3 = 0.05, p4 = 0.2, p5 = 0.3. k2 k1 k5
i depthT(ki) depthT(ki)·pi
1.10
Therefore, E[search cost] = 2.10.
This tree turns out to be optimal for this set of keys.
Comp 122, Spring 2004

23. Example Observations: Build by exhaustive checking?
Optimal BST may not have smallest height.
Optimal BST may not have highest-probability key at root.
Build by exhaustive checking?
Construct each n-node BST.
For each, assign keys and compute expected search cost.
But there are (4n/n3/2) different BSTs with n nodes.
Comp 122, Spring 2004

24. Optimal Substructure
Any subtree of a BST contains keys in a contiguous range ki, ..., kj for some 1 ≤ i ≤ j ≤ n.
If T is an optimal BST and T contains subtree T with keys ki, ... ,kj , then T must be an optimal BST for keys ki, ..., kj.
Proof: Cut and paste. T T
Comp 122, Spring 2004

25. Optimal Substructure
One of the keys in ki, …,kj, say kr, where i ≤ r ≤ j, must be the root of an optimal subtree for these keys.
Left subtree of kr contains ki,...,kr1.
Right subtree of kr contains kr+1, ...,kj.
To find an optimal BST:
Examine all candidate roots kr , for i ≤ r ≤ j
Determine all optimal BSTs containing ki,...,kr1 and containing kr+1,...,kj kr ki
kr-1
kr+1 kj
Comp 122, Spring 2004

26. Recursive Solution
Find optimal BST for ki,...,kj, where i ≥ 1, j ≤ n, j ≥ i1. When j = i1, the tree is empty.
Define e[i, j ] = expected search cost of optimal BST for ki,...,kj.
If j = i1, then e[i, j ] = 0.
If j ≥ i,
Select a root kr, for some i ≤ r ≤ j .
Recursively make an optimal BSTs
for ki,..,kr1 as the left subtree, and
for kr+1,..,kj as the right subtree.
Comp 122, Spring 2004

27. Recursive Solution When the OPT subtree becomes a subtree of a node:
Depth of every node in OPT subtree goes up by 1.
Expected search cost increases by
If kr is the root of an optimal BST for ki,..,kj :
e[i, j ] = pr + (e[i, r1] + w(i, r1))+(e[r+1, j] + w(r+1, j))
= e[i, r1] + e[r+1, j] + w(i, j).
But, we don’t know kr. Hence,
from (15.16)
(because w(i, j)=w(i,r1) + pr + w(r + 1, j))
Comp 122, Spring 2004

28. Computing an Optimal Solution
For each subproblem (i,j), store:
expected search cost in a table e[1 ..n+1 , 0 ..n]
Will use only entries e[i, j ], where j ≥ i1.
root[i, j ] = root of subtree with keys ki,..,kj, for 1 ≤ i ≤ j ≤ n.
w[1..n+1, 0..n] = sum of probabilities
w[i, i1] = 0 for 1 ≤ i ≤ n.
w[i, j ] = w[i, j-1] + pj for 1 ≤ i ≤ j ≤ n.
Comp 122, Spring 2004

29. Pseudo-code Time: O(n3) OPTIMAL-BST(p, q, n) for i ← 1 to n + 1
do e[i, i 1] ← 0
w[i, i 1] ← 0
for l ← 1 to n
do for i ← 1 to nl + 1
do j ←i + l1
e[i, j ]←∞
w[i, j ] ← w[i, j1] + pj
for r ←i to j
do t ← e[i, r1] + e[r + 1, j ] + w[i, j ]
if t < e[i, j ]
then e[i, j ] ← t
root[i, j ] ←r
return e and root
Consider all trees with l keys.
Fix the first key.
Fix the last key
Determine the root
of the optimal
(sub)tree
Time: O(n3)
Comp 122, Spring 2004

30. Elements of Dynamic Programming
Optimal substructure
Overlapping subproblems
Comp 122, Spring 2004

31. Optimal Substructure
Show that a solution to a problem consists of making a choice, which leaves one or more subproblems to solve.
Suppose that you are given this last choice that leads to an optimal solution.
Given this choice, determine which subproblems arise and how to characterize the resulting space of subproblems.
Show that the solutions to the subproblems used within the optimal solution must themselves be optimal. Usually use cut-and-paste.
Need to ensure that a wide enough range of choices and subproblems are considered.
Comp 122, Spring 2004

32. Optimal Substructure
Optimal substructure varies across problem domains:
1. How many subproblems are used in an optimal solution.
2. How many choices in determining which subproblem(s) to use.
Informally, running time depends on (# of subproblems overall)  (# of choices).
How many subproblems and choices do the examples considered contain?
Dynamic programming uses optimal substructure bottom up.
First find optimal solutions to subproblems.
Then choose which to use in optimal solution to the problem.
Comp 122, Spring 2004

33. Optimal Substucture
Does optimal substructure apply to all optimization problems? No.
Applies to determining the shortest path but NOT the longest simple path of an unweighted directed graph.
Why?
Shortest path has independent subproblems.
Solution to one subproblem does not affect solution to another subproblem of the same problem.
Subproblems are not independent in longest simple path.
Solution to one subproblem affects the solutions to other subproblems.
Example:
Comp 122, Spring 2004

34. Overlapping Subproblems
The space of subproblems must be “small”.
The total number of distinct subproblems is a polynomial in the input size.
A recursive algorithm is exponential because it solves the same problems repeatedly.
If divide-and-conquer is applicable, then each problem solved will be brand new.
Comp 122, Spring 2004